Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 3. Irreducible representations of SU(3), continued."

Time: 09:30

Room: MC 106

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Analysis Seminar

Speaker: Rahim Moosa (Waterloo)

"Real-analytic versus complex-analytic families of complex-analytic sets"

Time: 15:30

Room: MC 107

Suppose M is a compact complex manifold. Model theory (a branch of mathematical logic) provides at least two approaches to the study of the complex-analytic subsets of Cartesian powers of M, roughly corresponding to whether one focuses on the real or complex structure on M. We can view M as definable in the structure R_an; that is, as a real globally subanalytic manifold. On the other hand, we can work in the Zariski-type structure CCM where M is the universe and there are predicates for all complex-analytic subvarieties of Cartesian powers of M. The two approaches lead to different notions of a "definable family" of complex-analytic subsets. I will give a geometric characterization, obtained in joint work with Sergei Starchenko in 2008, of when these two notions coincide, in terms of the Barlet or Douady spaces. As a consequence one has that for M Kaehler the two notions coincide.

Graduate Seminar

Speaker: Richard Gonzales (Western)

"The equivariant Chern character"

Time: 16:30

Room: MC 107

A classical result of Atiyah and Hirzebruch establishes a deep connection between K-theory and cohomology, via the Chern character. The purpose of my talk is to describe this relation in precise terms, and give an overview of its generalizations to the equivariant setting. Along the way we introduce a new class of objects, coming from algebraic geometry, on which many of these classical topological techniques could be successfully applied.

Noncommutative Geometry

Speaker: Arash Pourkia (Western)

"Cyclic Cohomology 4"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Algebra Seminar

Speaker: Marc Moreno Maza (Western)

"Triangular decomposition of semi-algebraic systems"

Time: 14:30

Room: MC 107

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. We propose adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems.

We show that any such system can be decomposed into finitely many so-called "regular semi-algebraic systems". We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time with respect to the number of variables. We have implemented our algorithms and the experimental results illustrate their effectiveness. A software demonstration will conclude this talk.

This is a joint work with Changbo Chen (UWO), James H. Davenport (Bath U.), John P. May (Maplesoft), Bican Xia (Peking U.) and Xiao Rong (UWO).

The corresponding article is published in the Proceedings of the 2010 International Symposium of Symbolic and Algebraic Computation (ISSAC'10) and available at ww.csd.uwo.ca/~moreno/Publications/118_paper.pdf

Geometry and Topology

Speaker: Bjorn Dundas (Bergen)

"Two vector bundles and the splitting of the Dirac monopole over the three sphere"

Time: 15:30

Room: MC 107

(joint with Ausoni, Baas, Richter and Rognes)

Two vector bundles give rise to a geometrically defined cohomology theory extrapolating past the theory of vector bundles (K-theory) and differential forms (de Rham cohomology), capturing information related to cobordisms of manifolds beyond K-theory and deRham cohomology's reach.

The analytic and differential geometric understanding of two vector bundles is still very much in its infancy. There was a hope that an "integration of determinants through loops" construction would give an integral functor from two vector bundles to quantum field theories. However, the fact that the commutative ring spectrum representing complex K-theory does not support a determinant rules this out.

The first obstruction has a geometric interpretation: the one-dimensional two vector bundle represented by the Dirac monopole over the three sphere splits virtually.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Cyclic cohomology 5"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Algebra Seminar

Speaker: David Jeffrey (Western)

"Integration in computer algebra: problems, bugs and algebra"

Time: 14:30

Room: MC 107

Integration is sometimes said to be a solved problem in computer algebra, but integration problems are the source of a significant percentage of bug reports and complaints to Mathematica and Maple. The reasons for this are discussed and some remedies described.

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 4. Irreducible representations of SU(3), continued."

Time: 09:30

Room: MC 106

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Noncommutative Geometry

Speaker: Raphael Ponge (Tokyo)

"Noncommutative Geometry and Group Actions (first part)"

Time: 12:30

Room: MC 107

In many geometric situations we may encounter the action of a group G on a manifold M, e.g., in the context of foliations. If the action is free and proper, then the quotient M/G is a smooth manifold. However, in general the quotient M/G need not even be Hausdorff. Under these conditions how can we do diffeomorphism-invariant geometry?

Noncommutative geometry provides us with a solution by trading the badly behaved space M/G for a non-commutative algebra, which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah and Singer ultimately holds in the setting of noncommutative geometry. This enabled Connes and Moscovici to reformulation of the local index formula in the setting of diffeomorphism-invariant geometry.

The first part of the lectures will be a review of noncommutative geometry and Connes-Moscovici's index theorem in diffeomorphism-invariant geometry.

In the 2nd part, I will hint to on-going projects on the reformulation of the local index formula in two new geometric settings: biholomorphism-invariant geometry of strictly pseudo-convex domains and contactomorphism-invariant geometry of contact manifolds.

Geometry and Topology

Speaker: Matthias Franz (Western)

"Tensor products of homotopy Gerstenhaber algebras"

Time: 15:30

Room: MC 107

A Gerstenhaber algebra is a special kind of graded Poisson algebra. A homotopy Gerstenhaber algebra is a specific "up to homotopy" version of the former. Important examples of homotopy Gerstenhaber algebras are the Hochschild cochains of an associative algebra and the cochain complex of a simplicial set.

In this talk I will address the following problem: What structure does the tensor product of two homotopy Gerstenhaber algebras have? If time permits, I will also talk about formality results for homotopy Gerstenhaber algebras.

Noncommutative Geometry

Speaker: Raphael Ponge (Tokyo)

"Noncommutative Geometry and Group Actions (2nd part)"

Time: 13:00

Room: MC 107

In many geometric situations we may encounter the action of a group G on a manifold M, e.g., in the context of foliations. If the action is free and proper, then the quotient M/G is a smooth manifold. However, in general the quotient M/G need not even be Hausdorff. Under these conditions how can we do diffeomorphism-invariant geometry?

Noncommutative geometry provides us with a solution by trading the badly behaved space M/G for a non-commutative algebra, which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah and Singer ultimately holds in the setting of noncommutative geometry. This enabled Connes and Moscovici to reformulation of the local index formula in the setting of diffeomorphism-invariant geometry.

The first part of the lectures will be a review of noncommutative geometry and Connes-Moscovici's index theorem in diffeomorphism-invariant geometry.

In the 2nd part, I will hint to on-going projects on the reformulation of the local index formula in two new geometric settings: biholomorphism-invariant geometry of strictly pseudo-convex domains and contactomorphism-invariant geometry of contact manifolds.

Graduate Seminar

Speaker: Mehdi Mousavi (Western)

"The Riesz Approach to The Lebesgue Integral"

Time: 16:30

Room: MC 107

We define the Lebesgue integral directly by extending the Riemann integral. We will see that the passage of limit under the integral sign can be used effectively to define the Lebesgue integral, and avoids all the usual definitions of measures and measurable sets. This talk is also available to all undergrad students who have good backgroud in calculus.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Cyclic cohomology 6"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Symplectic Learning Seminar

Speaker: M. Mousavi (Western)

"Measure spaces and rearrangements"

Time: 13:00

Room: MC 105C

We will present results in measure theory that are needed in a generalization of the Shurn-Horn theorem to symplectomorphism groups of toric manifolds.

Algebra Seminar

Speaker: Dan Christensen (Western)

"Computation of traces in the representation theory of the symmetric and unitary groups"

Time: 14:30

Room: MC 107

I will review the classification of representations of the symmetric and unitary groups, and how they are related to each other. In particular, I will describe the Young projection operators whose images give the irreducible representations. Then I will give new formulas which use the Young projection operators to construct a family of orthogonal projections which are convenient for computations. Finally, I will describe how computations of traces of maps of symmetric group representations can be used to compute traces of maps of $U(n)$ representations for all $n$ at once. If I have time, I hope to package this up in the language of traced monoidal categories.

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 5. Irreducible representations of SU(3), continued."

Time: 10:00

Room: MC 107

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Cyclic cohomology 7, (K-theory for C^*-algebras (I): fundamentals of C*-algebras and basic K-groups)"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Symplectic Learning Seminar

Speaker: M. Mousavi (Western)

"Doubly Stochastic Operators"

Time: 13:00

Room: MC 105C

Doubly stochastic operators are analog of doubly stochastic matrices. We will define doubly stochastic operators by using a pre-order relation that can be defined by rearrangement of functions. We discuss different characterizations of doubly stochastic operators. If time permits we will explain a new representation for doubly stochastic operators.

Colloquium

Speaker: Roger Zierau (Oklahoma State)

"Differential operators on homogeneous spaces"

Time: 15:30

Room: MC 107

G-invariant differential operators D on a homogeneous space G/H will be discussed. A principle for constructing explicit solutions to Df=0 will be explained. This will be illustrated with several examples, such as Dirac operators.

Colloquium

Speaker: André Joyal (UQAM)

"TBA"

Time: 15:30

Room: MC 108

Colloquium

Speaker: André Joyal (UQAM)

"TBA"

Time: 15:30

Room: MC 107

Geometry and Topology

Speaker: Bert Guillou (Univ. of Illinois)

"cancelled"

Time: 15:30

Room: MC 107

Symplectic Learning Seminar

Speaker: M. Mousavi (Western)

"Schur-Horn-Kostant theorem for Symplectomorphisms of toric manifolds"

Time: 13:00

Room: MC 105C

We will start to prove the Orbit theorem, Schur's theorem, Horn's theorem and finally the Convexity theorem for symplectomorphism groups of toric manifolds. The main technique will be analogues of the diagonalization and spectral theorems.

Graduate Seminar

Speaker: Kavita Sutar (Northeastern University)

"Representations of quivers and some related geometry"

Time: 16:30

Room: MC 107

Cancelled.

Noncommutative Geometry

Speaker: Arash Pourkia (Western)

"Cyclic cohomology 8 (Hochschild homology of group algebra)"

Time: 14:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Symplectic Learning Seminar

Speaker: M. VanHoof (Western)

"Connectedness of Symp($CP^2$)"

Time: 13:00

Room: MC 104

In a recent paper, McDuff proves the connectedness of the symplectomorphism group of a resolution of a weighted projective space. Her argument is likely to adapt to some other orbifolds. As a stepping stone, we will explain the proof of the connectedness of Symp($CP^2$), a result originally proved by Gromov.

Colloquium

Speaker: Man Wah Wong (York)

"Laplacians related to the Heisenberg group"

Time: 15:30

Room: MC 107

We begin with the sub-Laplacian on the Heisenberg group. The twisted Laplacian is then introduced by taking the inverse Fourier transform of the sub-Laplacian with respect to the center of the Heisenberg group. After a recapitulation of the spectral theory of the twisted Laplacian in terms of the Wigner transform, the spectral theory and number theory of the twisted bi-Laplacian obtained by Gramchev, Pilipović, Rodino and me are reported. We end the talk with a glimpse into a connection of the twisted bi-Laplacian with the Riemann zeta-function.

Algebra Seminar

Speaker: Stephen Watt (Western)

"The mathematics of mathematical handwriting recognition"

Time: 14:30

Room: MC 107

Accurate computer recognition of handwritten mathematics offers to provide a natural interface for mathematical computing, document creation and collaboration. Mathematical handwriting, however, provides a number of challenges beyond what is required for the recognition of handwritten natural languages. For example, it is usual to use symbols from a range of different alphabets and there are many similar-looking symbols. We present a geometric theory that we have found useful for recognizing mathematical symbols. Characters are represented as parametric curves approximated by certain truncated orthogonal series. This maps symbols to the low-dimensional vector space of series coefficients. The beauty of this theory is that a single, coherent view provides several related geometric techniques that give a high recognition rate and do not rely on peculiarities of the symbol set.